Normal Distribution Calculator
Enter a mean and standard deviation, then find the probability below, above, between or outside values of x — or work backwards from a probability to the cut-off value and the central interval. Every answer comes with its z-scores, the working and a shaded bell curve. Free, no sign-up.
Results
0.8413
P(X < 115)
84.13%
As a percentage
1
z-score
How this was calculated
z = (x − μ) / σ = (115 − 100) / 15 = 1
P(X < 115) = Φ(1) = 0.8413
Normal distribution, μ = 100, σ = 15. The shaded area is the probability.
How to Use the Normal Distribution Calculator
- 1. Enter the mean and standard deviation. Use the values μ and σ for the variable you are modelling — 100 and 15 for IQ scores, for example.
- 2. Choose what to find. A probability, for the region below, above, between or outside values; or a value, when you already have a probability.
- 3. Enter x, a and b, or the probability. Values can be any real numbers; a probability must lie strictly between 0 and 1.
- 4. Read the answer. You get the probability (also as a percentage) with its z-scores and working — or the cut-off values — plus a bell curve with the area shaded.
How Normal Probabilities Are Calculated
A normal distribution is described completely by its mean μ and standard deviation σ. To find an area, the calculator standardises each value with z = (x − μ) / σ and reads Φ(z), the standard normal cumulative distribution function — the area to the left of z. Every other region follows from that one function.
| Question | Formula |
|---|---|
| P(X < x) | Φ(z) |
| P(X > x) | 1 − Φ(z) |
| P(a < X < b) | Φ(zb) − Φ(za) |
| P(X < a) + P(X > b) | Φ(za) + 1 − Φ(zb) |
| x with P(X < x) = p | x = μ + σ Φ−1(p) |
| Central interval holding p | μ ± σ Φ−1((1 + p) / 2) |
Φ has no closed form. The calculator evaluates it from the complementary error function to full double precision, so its answers match statistical software to every digit shown, with no table rounding. For a continuous distribution P(X < x) and P(X ≤ x) are equal, because a single point has zero probability.
Use this calculator when you know μ and σ and want an area or a cut-off. To standardise a single score — how many standard deviations it sits from the mean — the Z-Score Calculator does exactly that, and the Z-Table shows the standard normal areas as a printed table.
Worked Example: IQ Scores
IQ scores are scaled to a mean of 100 and a standard deviation of 15. What share of people score below 115? The z-score is (115 − 100) / 15 = 1 and Φ(1) = 0.8413, so 84.13% score below 115 and 15.87% above.
Between 85 and 115 the z-scores are −1 and 1, so P(85 < X < 115) = Φ(1) − Φ(−1) = 0.6827: about two-thirds of people lie within one standard deviation of the mean. Above 130 (z = 2) the area is 0.0228, roughly one person in 44.
Working backwards with p = 0.95, P(X < x) = 0.95 gives x = 124.6728 (z = 1.6449), so only 5% score above about 124.7. The central 95% of scores runs from 70.6005 to 129.3995, which is μ ± 1.96σ.
The 68–95–99.7 rule
About 68.27% of any normal distribution lies within one standard deviation of the mean, 95.45% within two and 99.73% within three. It is a quick check on any answer: a value 2σ above the mean should leave about 2.3% in the upper tail.
Common mistakes
- Entering the variance as the SD. Type σ, not σ². With 225 in the SD box instead of 15, every z-score shrinks fifteen-fold and every tail area is wrong.
- Assuming the data are normal. Incomes, reaction times and many counts are skewed, and normal areas for them can be badly wrong, especially in the tails.
- Mixing up the tails. P(X > x) is 1 minus the cumulative probability, not the cumulative probability itself — the most common slip when reading a z-table.
Frequently Asked Questions
- How do I find the probability between two values in a normal distribution?
- Choose "Between a and b" and enter both values. The calculator standardises each one and returns Φ(zb) − Φ(za). For IQ scores with μ = 100 and σ = 15, P(85 < X < 115) = 0.6827.
- What is the inverse normal distribution?
- It runs the calculation backwards: given a probability p, it finds the value x with P(X < x) = p, using the quantile function x = μ + σ Φ−1(p). Use it for cut-offs such as the 90th percentile or the score needed to land in the top 5%.
- How is this different from a z-score calculator?
- A z-score calculator standardises one value and reports its percentile. This calculator answers questions about regions of the distribution — below, above, between or outside — and inverse questions, for any mean and SD, and draws the area on the curve.
- Is P(X < x) the same as P(X ≤ x)?
- Yes, for the normal and every other continuous distribution: a single exact value has probability zero, so including or excluding the endpoint changes nothing. That is not true for discrete distributions such as the binomial, where P(X ≤ 3) and P(X < 3) differ by P(X = 3).
- How accurate are the results?
- Cumulative probabilities come from the complementary error function and the inverse from a rational approximation refined with a Halley step, both accurate to about 15 significant digits — far beyond the four decimals of a printed z-table. Very small tail areas are shown in scientific notation rather than rounded to zero.
- What does the standard deviation do to the bell curve?
- It sets the spread. A larger σ makes the curve lower and wider, a smaller σ taller and narrower, but the total area is always 1. That is why the same z-score always cuts off the same area, whatever the mean and SD.